die Beteiligung
participation
Active participation or involvement in an activity, event, or process.
Die Beteiligung der Bürger an dem Projekt war sehr hoch.
Citizen participation in the project was very high.

participation / stake
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participation
Active participation or involvement in an activity, event, or process.
Die Beteiligung der Bürger an dem Projekt war sehr hoch.
Citizen participation in the project was very high.

'Die Beteiligung' (participation/stake) is derived from the verb 'sich beteiligen' (to participate), which in turn goes back to the noun 'der Teil' (part). It describes the action of becoming a 'Teil' (part) of something or having a 'Teil' (share) in something.
Alle können sich an der Diskussion beteiligen.
Everyone can participate in the discussion.
To say what someone participates in or what someone owns a share in, the preposition 'an' with the dative is almost always used. This structure applies to both main meanings of the word.
Die Beteiligung an der Wahl war dieses Jahr sehr niedrig.
Participation in the election was very low this year.
You can tell from the context whether participation or a financial stake is meant. If it is about projects, elections, or discussions, participation is meant. If it is about companies, investors, and percentages, a financial stake is meant.
Wir freuen uns über die rege Beteiligung an unserem Sommerfest.
We are pleased about the lively participation in our summer festival.
When it is about participation, 'Beteiligung' is often combined with adjectives such as 'aktiv' (active), 'rege' (lively), 'hoch' (high), or 'gering' (low) to describe the extent of involvement.
Eine aktive Beteiligung der Eltern ist in der Schule sehr erwünscht.
Active participation by parents is very welcome at the school.
share / proportion
'der Anteil' and 'die Beteiligung' can both express that someone owns a share of something, especially in companies. 'Anteil' emphasizes the part itself, that is, how large the ownership share is. 'Beteiligung' tends to emphasize that someone has a financial stake in a company; it often sounds like a business connection. They can be confused because a Beteiligung usually consists of an Anteil. In practice: use Anteil when you mean the part or its size. Use Beteiligung when you mean the economic connection or investment.

Die Firma plant eine Beteiligung an einem jungen Technikunternehmen.
The company is planning to take a stake in a young technology company.
Ihr Anteil am Gewinn wird am Ende des Jahres berechnet.
Her share of the profit will be calculated at the end of the year.